First of all, recall that the inequality \(\mathfrak {S}_Y\ge \mathfrak {I}_Y\)  holds, where \(\mathfrak {I}_Y\)  is the isoperimetric constant of \(M\)  [1]}: this is the infimum of the quantity
\(\frac{A(N)^3}{\min (V(M_1),V(M_2))^2}\) 
